Equimatchable Graphs on Surfaces

A graph G is equimatchable if any matching of G is a subset of a maximum-size matching. From a general description of equimatchable graphs in terms of GallaiEdmonds decomposition [Lesk, Plummer, and Pulleyblank, "Equimatchable graphs", Graphs Theory and Combinatorics, Academic press, London, (1984) 239-254.] it follows that any 2-connected equimatchable graph is either bipartite or factor-critical. In both cases, the Gallai-Edmonds decomposition gives no additional information about the structure of such graphs. It is well known that for any vertex v of a factor-critical equimatchable graph G and a minimal matching Mv that isolates v the components of the graph G \ (Mv ∪ {v}) are all either complete or regular complete bipartite. We prove that for any 2-connected factor-critical equimatchable graph G, the graph G \ (Mv ∪ {v}) has at most one component, and use this result to establish that the maximum size of such graphs embeddable in the orientable surface of genus g is Θ(g), improving on previous bound O(g3/2). In addition, we bound the maximum size of k-degenerate 2-connected factor-critical graph. Moreover, for any non-negative integers g and k we provide a construction of arbitrarily large 2-connected equimatchable bipartite graphs with orientable genus g and a genus embedding with face-width k. The structure of factor-critical equimatchable graphs with a cut-vertex or a 2-cut was determined in [J. Graph Theory, 10(4):439–448, 1986.]. We extend these results and for all k ≥ 3 we describe the structure of factor-critical equimatchable graphs with a k-vertex-cut. More precisely, for every k ≥ 3, we prove that if a k-connected equimatchable factor-critical graph G has at least 2k+ 3 vertices and a k-cut S such that G− S has two components with sizes at least 3, then G−S has exactly two components and both are complete graphs. Consequently, if k ≥ 4 such graphs have independence number 2. Additionally, we provide also a characterisation of k-connected equimatchable factor-critical graphs with a k-cut S such that G−S has a component with size at least k and a component with size 1 or 2. Finally, we show that if every minimum cut S of an equimatchable factor-critical graph separates a component with size at most two, then the independence number can be arbitrarily high and the number of components can be as high as |V (S)|.